On the Extension of the Three-Term Recurrence Relation to Probabilities Distributions without Moments

In this paper, we extend the three-term recurrence relation for orthogonal polynomials associated with a probability distribution having a finite moment of all orders to a class of orthogonal functions associated with an infinitely divisible probability distribution μ having a finite moments of order less or equal to four. An explicit expression of these functions will be given in term of the Lévy-Khintchine function of the measure μ .


Introduction
It has been known from [1] and [2] that for every probability distribution µ with finite moments of all orders, there exits a family of monic orthogonal polynomials n P and a paire of sequences 0 1 , , α α  and 0 1 2 1, , , 0 w w w = >  satisfying the three-term recurrence relation (or the tri-diagonal Jacobi relation) { } The sequences ( n α ) and ( n w ) are called the Szego-Jacobi parameters of µ .
operators.This allows to associate, in a canonical way, to any random variable with all moments commutation relations that generalize the Heisenberg commutation relations (corresponding to the Gauss-Poisson class).From the mathematical point of view, this approach has led to some new results in the theory of OP.
In order to give this operator interpretation, we shall recall the notion of the interacting Fock probability space associated with the measure µ (See [3] for more details).
Consider an infinite-dimensional separable Hilbert space  , in which a complete orthonormal basis { } ; n n Φ ∈ is chosen.Let 0 ⊂   denote the dense subspace spanned by the complete orthonormal basis { } n Φ .
Given the sequence { } , n w n∈  , we associate linear operators ( ) given by: be the space of classes of complex valued, square integrable functions w.r.t µ .In the following, we simply denote it by ( )

2
L µ and we assume that the sub-space Then the U is unitary and we have This means that the field operator : the position operator : x q M = on ( )

2
L µ providing, in this way, a new interpretation of the recursion relation driving by OP in term of CAP operators.
Since the random variable with distribution µ can be identified, up to stochastic equivalence ≡ , with the position operator q on ( ) This result motivated the apparition of a series of papers [4]- [9] dealing in the same context and provided many applications in the theory of quantum probability.In the paper [4], a similar result was obtained but for the family of operator q is the field operator where q is the function . See papers [10] and [11] in which the operator Q was widely studied.
In this approach, the construction was not based on the orthogonal The main result of this paper will be given in Section 4, so that we compute the action of the position operator x q M = on the orthogonal functions , n E α .This provide such a generalization of the tri-diagonal recursion relation for OP.Finally, the explicit form of theses functions will be given.

The Bosonic Fock Space
Let  be a separable Hilbert space.Let us denote tensor product of n-copies of  (resp.f ∈  ) and let u σ be the unique unitary operator such that : , where n σ ∈S is a permutation of n-variables.
, were Φ is the vacuum vector, let be the orthogonal projection.
We define Let us denote ( ) The bosonic creation and annihilation operators are defined, on the total set as follows: : and ( ) and where .ˆ denotes omission of the corresponding variable.The preservation operator associated with the self adjoint operator T on  is given by: :

The Quantum Decomposition of Classical Random Variables with I.D-Distributions
In this section, we recall briefly, what has been obtained in the paper [4] around quantum decomposition of random variables with I.D-distributions and having a finite second order moment.
Let us consider a random variable X with I.D-probability distribution µ having a finite second order moment.It is known (see [13]), that the Fourier transform of µ given by ( ) where Ψ is given by such that , γ σ ∈  and ν is the the Lévy measure of µ .The function Ψ is called the Lévy-Khintchine function or the characteristic exponent associated with µ .Since the second order moment of µ is finite, the same result will be true for ν , i.e,: We suppose also that the gaussian part of µ is null (i.e., 0 σ = ).Under these conditions, we have the following results: The family { } of the trigonometric functions is total in ( )

2
L µ and the family of the functions Then by applying the Araki-Woods-Parthasarathy-Schmidt isomorphism in [12] for the infinitely divisible positive definite kernel , : e , , , we have proved the following theorem (See [4] for more details and descriptions).
Theorem 2.1.The unique linear operator U given on the exponential vectors : is an unitary isomorphism from the Fock space ( ) ( ) Let q be the multiplication (position) operator in ( ) where U is the isomorphism defined by (12).Since µ is a finite measure on  , the operator q is self-adjoint (see [14]  Then the generalized field operator Q takes the form

Notations
We denote by τ the set of all sequences of non negatives integers with finite number of nonzero entries.In the sequel n  (resp.n  ) will be interpreted as subset of the set τ (resp. ( ) 2 l  ).Throughout the remain of this paper we shall use the following notations: , , , , , , , , 0, 0, The support of such element α τ ∈ is defined by , , , ,  : In particular if ( ) 1 , , s i i π α =  and n α = , so that α takes the form ( ) 0, , 0, , 0, , 0, , 0, , 0, , 0, 0, with , From [15], we recall the following identity which is the analogue of the multinomial Newton formula ( ) which take place whenever the series g ≥ is an Hilbertian basis of it, then the set ( ) ( )  be the canonic basis of ( ) and .
, then k α − can be defined as in ( 16), however it is not an element of τ , because its k th -entry ( ) In this case, we adapt by convention that Finally, we recall that

Computation of the Action of the Generalized Field Operator on the Basis (Φn)n
In the remain, we take ( ) and we assume that second order moment of ν is finite.Let 1 g be the function given by ( ) ( ) ( ) is also total.Then by the Gram-Schmidt procedure, we construct an Hilbertian basis of it, that is denoted by , , , , Then ( ) then it is sufficient to prove that ( ) We have where we have used the condition (10).
From ( 5), we have ( ) Here, we have two cases: If ( ) ) Since ( ) , then it can be written as follows: Using the fact that n S is bounded, the Equation (26) becomes ( ) But we have for ( ) The action of the generalized field operator Q on the basis We assume that µ is continuous w.r.t the Lebesgue measure with Radon-Nikodym derivative ρ .Then for all n ∈  and α τ ∈ , one has where, ( ) ( ) ( ) where the series converge in ( )

2
L ν .It follows, from the multinomial Newton formula (14), that  From the definition of U, we get Its known that B ± are mutually adjoint and the linear subspace 0 interacting Fock probability space associated with µ .The operators B + and B − are called the creation operator and the annihilation operators respectively.The linear operator given by

2 L 2 L µ . Then the 1 U
random variables having an infinitely divisible distribution (I.D-distribution in the following) and having only the moment of the second order.Here, similarity means that the quantum decomposition can be obtained also for this family of random variables.Based on the notion of the positive definite kernel and using the Lévy-Khintchine function established for the I.D-distributions, the paper [4] constructed a natural isomorphism U from the Fock space -space w.r.t the Lévy measure ν to the space ( ) − -image of the position

2 L 2 L
polynomials sequence associated with µ .But it required only the infinite divisibility property, where the Lévy-Khinchine function have played an important role.Then one can ask about the analytic form of the relation (4), or equivalently the counterpart of the three-term recurrence relation.The only obscure point is the existence of such an analogue of the sequence of the orthogonal polynomials.Since the hypothesis on moments is not satisfied, such a sequence of orthogonal polynomial does exist.But the isomorphism U provided us a such chaos-decomposition of the space ( ) µ .For this reason we ask the question if there exist a such analogue for the family of orthogonal polynomial, if it is the case it must be a total family of orthogonal functions in the space ( ) µ satisfying a recursion relation similar to the well known for OP.

)
gives that the relation (25) sill true.Hence (22) is proved.Now, it remains to justify (23).From(7), we get 27) Journal of Applied Mathematics and Physics  denotes the Fourier transform on ( ) The operator Q is called the generalized field operator.
y Exp f y ∈  is in the domain of Q.Moreover, one has the following theorem: Theorem 2.2.Let q be the function given by